A new comprehensive analysis of growth dynamics of an ensemble of gas bubbles on the nucleation stage at degassing in a gas-supersaturated liquid solution is presented. The Laplace pressure in supercritical bubbles is strictly taken into account in the analysis based on the mean-field approach to nucleation kinetics. The whole distribution of supercritical bubbles in sizes, gas supersaturation in the liquid solution at any time at the nucleation stage as well as the total duration of the nucleation stage have been described analytically. The scaled universal forms independent of specific parameters of the gas and solvent components have been found for the maximal bubble radius and the reduction of mean gas supersaturation as functions of time. It has been shown that the Laplace pressure in the supercritical bubbles considerably affects the bubble distribution at the beginning of nucleation stage but becomes less important to the end of the nucleation stage.